It is well known to apply electromagnetic radiation to evaluate optical properties of sample systems. The disclosed invention can comprise any means for generating an electromagnetic beam and causing it to impinge upon a sample system, such as:                reflectometer;        spectrophotometer;        ellipsometer;        spectroscopic ellipsometer;        polarimeter; and        spectroscopic polarimeter;with reflected or transmitted electromagnetic radiation being directed to enter a detector.        
For breadth it is noted that while semiconductor sample systems with one or more thin film on their surface are typical of those to which ellipsometry can be applied, essentially any sample can be subjected to investigation using electromagnetic radiation. Non-limiting Examples are amples which comprise:                Materials with High or Low Extinction Coefficient (K);        Materials with High or Low Refractive Index (N);        Metals;        Semimetals;        Semiconductors;        Insulators;        Transparent Oxides;        Liquids;        Fluids;        oils;        Lubricants;        Biological materials;        Nucleic Acids;        DNA &/or RNA;        Proteins;        Amino Acids;        Carbohydrates;        Waxes;        Fats;        Lipids;        Plant material;        Animal material;        Fungi material;        Microbe material;        Tissues;        Condensates;        Combination Solid and Liquid &/or Gas sample systems;        Liquid Crystals;        free standing films;        Porous materials;        Alloys;        Compounds;        Composites;        Ceramics;        Polymers;        Fiberous materials;        Wood containing materials;        Paper containing materials;        Plastics;        Crystaline materials;        Amorphous materials;        Polycrystaline Materials;        Glassy materials;        Homogeneous materials;        Inhomogeneous materials;        Superlattices;        Superconductors;        Lamgmuir-Blodgett materials;        Monolayers;        Fractional Monolayers;        Multi layers;        Samples comprising Quantum Dots &/or Wells;        Polymers;        Conjugated Polymers;        Films of any material on substrate of another material;        Monoparticles;        Composites containing monoparticles;        Nanomaterials;        Materials containing Nanomaterials;        Superlattices with Nanoparticles;        combinations of the above.        
It is further noted that many types of ellipsometer systems exist. Examples are Nulling, Rotating Polarizer, Rotating Analyzer, Rotating Compensator, Modulation Element etc. but all have in common that they provide data which allow determination of PSI (Ψ) and DELTA (Δ) of a Sample, where (Ψ) and (Δ) are defined by:ρ=rp/rs=Tan(Ψ) exp(iΔ)Tan (Ψ) or Cos (Δ), coordinates on a Poincare Sphere or other descriptors of the Polarization State of electromagnetic radiation are to be understood as equivalents to PSI (Ψ) and DELTA (Δ), as are the Coefficients “IN” “S” & “C”:N=Cos(2Ψ);C=Sin(2Ψ)Cos(Δ);S=Sin(2Ψ)Sin(Δ);in Modulation Ellipsometers, (see U.S. Pat. Nos. 5,416,588 and 5,956,145 which are incorporated by reference hereinto).
Further, evaluation can be achieved for at least one sample system characterizing selection from the group consisting:                energy gap;        index of refraction;        growth rate;        etch rate;        thickness;        extinction coefficient;        carrier concentration;        alloy ratio;        critical point;        depolarization rate;        inhomogenuity;        grading;        anisotropy;        temperature;        crystalinity;        stress;        strain;        surface layer roughness;        interface layer;        interface roughness;        electro-optic coefficient;        magneto-optic coefficient;        chemical bond presence;        chemical bond strength;        sample system phase;        combinations of the above.        
It is also noted that data can be obtained which is characterized by at least one selection from the group consisting of:                reflected electromagnetic radiation;        transmitted electromagnetic radiation;        single sample system invesitgated;        multiple sample systems simultaneously        investigated;        single angle of incidence of electromagnetic        radiation;        single sample system orientation;        multiple sample system orientations;        multiple angles of incidence of        electromagnetic radiation;        date acquired from single instrument;        data acquired from multiple instruments;        focused beam of electromagnetic radiation;        divergent beam of electromagnetic radiation;        unfocused beam of electromagnetic radiation;        in-situ;        ex-situ;        electro-optic;        electro-opticcoincident with ellipsometric data;        kerr magneto-optic;        kerr magneto-optic coincident with ellipsometric data;        combinations of the above.        
Briefly, methodology provides that electromagnetic radiation be caused to interact with a sample system, with detected changes in intensity and/or polarization state being used to evaluate the sample system optical constants. Spectrophotometry and Reflectometry detect changes in Intensity of a beam of electromagnetic radiation resulting from interaction with a sample system. Ellipsometry characterizes change in polarization state caused by interaction of a beam of electromagnetic radiation with a sample system. And while strong correlation typically exists between thin film thickness and optical constants, Ellipsometric data can, at times, be sufficient to evaluate the thickness of a thin flim on a sample system and its refractive index and extinction coeficient, (or the mathematically equivalent real and imaginary parts of the dielectric function). Combined with Spectrophotometry and Reflectometry data serves to to enable less correlated determination of sample system properties.
Further, while use of data at a single wavelength can provide beneficial results in many situations, it is typical to obtain data at a multiplicity of wavelengths, and perhaps at a plurality of Angles of Incidence (AOI) at which a beam of electromagnetic radiation is caused to approach a sample system. This allows determining the refractive index and extinction coeficient or the mathematically equivalent real and imaginary parts of the dielectric function over a spectroscopic range.
Continuing, it is known to:                provide a sample system and obtain characterizing data therefrom comprising change in spectroscopic electromagnetic radiation caused to interact therewith, said change being caused by said interaction, and        provide a mathematical model of the sample system; and        perform a regression procudure of the mathematical model onto the data to evaluate parameters in the mathematical model.Where data is to be obtained in two different wavelength ranges, possible non-limiting combination of instruments which operate in said different spectral ranges are:        Two Ellipsometer Systems operating in different wavelength ranges;        One Ellipsometer and a Transmission Spectrometer;        One Ellipsometer and a Reflection Spectrometer;        Two Spectrometer Systems, one Reflection mode and one Transmission mode.        
Again, where ellipsometry is applied the change is in polarization state, rather than intensity of the spectroscopic electromagnetic radiation caused by reflection from or transmission through the sample system.
By said known methodology insight to the composition and optical properties of the sample system can be determined.
Typically the spectroscopic electromagnetic radiation is obtained using an instrument which provides wavelengths in a range selected from the group:                Radio;        Micro        Far Infrared (FIR);        Infrared (IR);        NIR-VIS-NUV;        Near Infrared (NIR);        Visual (VIS);        Near Ultraviolet (NUV);        Ultraviolet (UV);        Deep Ultraviolet (DUV);        Vacuum Ultraviolet (VUV);        Extreme Ultraviolet (EUV);        X-Ray (XRAY);and while wavelengths from one said selection is often sufficient to provide good evaluation of parameters in a mathematical model of a sample system, it does occur that some parameters in a mathematical model can not be investigated by wavelengths in a single selected range of wavelengths. For instance, wavelengths in the IR range are typically appropriate for investigation of the nature of chemical bonds in molecularly bonded sample systems, and IR, NIR, VIS and UV wavelengths are all suited to obtaining information about absorption and electron transitions in numerous material, (eg. semiconductors, semi-metals and metals). NIR wavelengths are particularly applicable to investigtion of semi-metals and VIS wavelengths to metals. Another example is that the angle of incidence of an IR beam with respect to a normal to a surface of a sample system is often strongly correlated with the absolute index of refraction of the sample system, but UV-VIS-NIR wavelengths are often capable of accurately evaluating the index and its dispersion, thereby enabling breaking of said correlation.        
The following provides insight to various Sample System Physical Parameters and Spectral Ranges typically utilized to provide insight thereto:
PHYSICAL PARAMETERSPECTRAL RANGEPHONON ABSORPTIONMID-IR (MIR) TO FAR-IR (FIR)MOLECULAR BOND ABSORPTIONMIR TO FIRCHEMISTRY INFORMATION)FREE CARRIER ABSORPTIONUV-VIS AND LONGER WAVELENGTHSIN METALSFREE CARRIER ABSORPTIONNEAR-IR (NIR)IN CONDUCTIVE OXIDESFREE CARRIER ABSORPTIONMIR TO FIRIN SEMICONDUCTORSSURFACE ROUGHNESSUV-VIS (THOUGH DIFFERENTWAVELENGTH RANGES GIVEINFORMATION ON DIFFERENTLEVELS OF ROUGHNSS)SMALL SCALE INDEX GRADINGABSORBING REGIONS IN VACUUM ULTRAVIOLET (VUV) TO NIRLARGE SCALE INDEX GRADINGANY SPECTRAL REGIONTHICKNESS NON-UNIFORMITYANY SPECTRALALTHOUGH RANGE SENSITIVITYINCREASES AT SHORTER WAVELENGTHSANISOTROPYANY SPECTRAL RANGE - VARIES WITHSAMPLEDICHROISMABSORBING REGION-TYPICALLY UV ORIRABSORPTION DUE TO ELECTRONICUV TO VIS AND NIR - NIR WITHTRANSITIONSNARROW BANDGAP SEMICONDUCTORSALLOY RATIOSUV-VISNATIVE OXIDESUV-VIS(SEMICONDUCTORS)ULTRA-THIN FILM THICKNESSUVOXYGEN IN LIVING TISSUES &VIS RED-NIRWATER IN VARIOUS STATES &BIOLOGICAL MATTER
It is also noted that published Optical Constant data for many materials, (eg gold), even today often have discontinuous steps or jumps therein which result from the use of different investigating systems on either-side of the discontinuity. That is, the Optical Constants arrived at using data obtained from an Ellipsometer capable in the IR wavelength range, and Optical Constants arrived at using data obtained from an Ellipsometer capable in the NIR-VIS waelength range are often not continuous at the interface wavelengths. The discontinuity is, of course, an artifact.
In view of the above it should be apparant that simultaneous use of information gathered from various wavelength ranges from the same or different investigation systems in a single regression procudure, would enable the ability to better evaluate sample systems and provide more realistic results.
While it is often the practice to regress on a point by point, (ie. wavelength by wavelength), basis in wavelength regions where a sample system is opague, regression based evaluation of parameters in a mathematical model preferably involves parameterization in wavelength regions where a sample system, (eg. a bulk material and substrates with layers thereupon), is transparent and where reflected electromagnetic radiation demonstrates minimal absorption and typically interference efects resulting from reflections from interfaces between layers as well as from the surface. Parameterization can provide Kramers- Kronig consistency, resulting in physical optical constants, and often reduces the number of parameters that are required to be evaluated to accurately model very irregular data. Parameterization also avoids discontinuities and reduces the effects of noise in optical constants by modeling with smooth mathematical functions. Disclosed in the J.A. Woollam Co. W-VASE32 (Registered Trademark), Manual are various approaches, including the mathematics thereof, to Parameterization, including:                Cauchy;        Cauchy +Urbach absorption;        Sellmeier Oscillator, (zero broadened);        Lorentz Oscillator;        Gaussian Oscillator;        Harmonic Oscillator;        Drude Oscillator;        Tauc-Lorentz Oscillator;        Cody-Lorentz Oscillator;        Tanguay;        Ionic Oscillator;        TOLO;        Gauss-Lorentz Oscillator;        Gauss-Lorentz Oscillator Asymetric Doublet (GLAD) Oscillator;        Herzinger-Johs Parametric Semiconductor Oscillator Model;        Psemi-Eo Oscillator;        Critical Point Parabolic band (CPPB);        Adachi Oscillator Model;        Pole;wherein the Oscillators are preferably Kramers-Kronig consistent.        
Following directly are mathematical descriptions of many of the foregoing Oscillator Structures, which are presently available in the J.A. Woollam Co. WVASE32 (Registered Trademark), GEMOSC™ Layer. Said GEMOSC™ layer can be invoked once Dielectric Function plots are known over a Spectroscopic Range by other means, and enables placing Oscillator Strurctures of appropriate shape at appropriate locations under, for instance, the imaginary part of said Dielectric Function such that summation of their contributions at each wavelength results in said Dielectric Function. The oscillator Structures are Kramers-Kronig consistant, hence modeling the Imaginary part allows calculation of the real part.
e1 Offset:Purely real constant added to ε1. It is equivalent to “εco”,which is often seen in scientific literature.Sellmeier:StyleEquationFit ParametersSell.0 (nm)Sell.5 (μm)      ɛ    n_sellmeier    =                              A          n                ·                  λ          2                    ⁢              λ        n        2                            λ        2            -              λ        n        2            Ampn = An (nm−2), Wvln = λn (nm)Ampn = An (μm−2), Wvln = λn (μm) Sell.1 (nm)Sell.6 (μm)      ɛ    n_sellmeier    =                              A          n                ·                  λ          2                    ⁢              λ        n                            λ        2            -              λ        n        2            Ampn = An (nm−1), Wvln = λn (nm)Ampn = An (μm−1), Wvln = λn (μm) Sell.2 (nm)Sell.7 (μm)      ɛ    n_sellmeier    =                    A        n            ·              λ        2                            λ        2            -              λ        n        2            Ampn = An (dimensionless),Wvln = λn (nm) or Wvln = λn (μm)n = 1, 2Cauchy:StyleEquationFit ParametersChy.0      N    n    =            A      n        +                  B        n                    λ        2              +                  C        n                    λ        4            Ann = An, Bnn = Bn, Cnn = Cn, (dimensionless)λ (μm)       K    n    =      αⅇ          β      ⁡              (                  1.24          ⁢                      (                                          1                λ                            -                              1                r                                      )                          )            Akn = α (dimensionless)Bkn = β (μm−1),       ɛ    n_Cauchy    =            (                        N          n                +                  iK          n                    )        2  Ckn = γ (μm) (user adjustable but not fittable.Only valid when λ > γ)DrudeStyleEquationFit ParametersDrd.0 (eV)Drd.5 (cm−1)      ɛ    n_Drd    =      -                            A          n                ⁢                  Br          n                                      E          2                +                              iBr            n                    ⁢          E                    Ampn = An (eV)Brn = Brn (eV)Ampn = An (cm−1),Brn = Brn (cm−1) Drd.1 (eV) Drd.6 (cm−1)      ɛ    n_Drd    =      -                  A        n                              E          2                +                              iBr            n                    ⁢          E                    Ampn = An (eV2),Brn = Brn (eV)Ampn = An (cm−2),Brn − Brn (cm−1)rho-tau Drude & N-mu Drude:CalculatedFitParametersStyleEquationParameters(mstar known)Rho-tau.0      ɛ    n_rtDrd    =            -              ℏ        2                            ɛ        0            ⁢                        ρ          n                ⁡                  (                                                    τ                n                            ·                              E                2                                      +                          i              ⁢                                                          ⁢              ℏ              ⁢                                                          ⁢              E                                )                    rhon = ρn (Ω-cm),taun = τn (10−15 sec) Rho-tau.5      ρ    n    =                    m        *                              N          n                ⁢                  q          2                ⁢                  τ          n                      =          1              q        ⁢                                  ⁢                  μ          n                ⁢                  N          n                    rhon = log10(ρn) (Ω-cm)taun = τn (10−15 sec)log10(N) (cm−3)μ (cm2V−1 sec−1) N-mu.0      ɛ    n_NμDrd    =            -              ℏ        2                            ɛ        0            ⁢                        ρ          n                ⁡                  (                                                    τ                n                            ·                              E                2                                      +                          i              ⁢                                                          ⁢              ℏ              ⁢                                                          ⁢              E                                )                    Nn = Nn (cm−3),mun = μn (cm2V−1 sec−1)N (cm−3),μ (cm2V−1 sec−1) N-mu.5      ρ    n    =                    m        *                              Nq          2                ⁢        τ              =          1              q        ⁢                                  ⁢                  μ          ⁢          N                    Rhon − log10(Nn) (Ω-cm)taun = τn (10−15 sec)log10(N) (cm−3)μ (cm2V−1 sec−1)Lorentz:StyleEquationFit ParametersLor.0 (eV) Lor.5 (cm−1)      ɛ    n_Lorentz    =                    A        n            ⁢              Br        n            ⁢              E        n                            E        n        2            -              E        2            -                        iBr          n                ⁢        E            Ampn = An (dimensionless),Enn = En (eV), Brn = Brn (eV)Ampn = An (dimensionless),En1 = En (cm−1), Br1 = Brn (cm−1) Lor.1 (eV) Lor.6 (cm−1)      ɛ    n_Lorentz    =                    A        n            ⁢              E        n                            E        n        2            -              E        2            -                        iBr          n                ⁢        E            Ampn = An (eV),Enn = En (eV), Brn = Brn (eV)Ampn = An (cm−1),Enn = En (cm−1), Brn = Brn (cm−1) Lor.2 (eV) Lor.7 (cm−1)      ɛ    n_Lorentz    =                    A        n            ⁢              E        n        2                            E        n        2            -              E        2            -                        iBr          n                ⁢        E            Ampn = An (dimensionless),Enn = En (eV), Brn = Brn (eV)Ampn = An (dimensionless),Enn = En (cm−1), Brn = Brn (cm−1)HarmonicStyleEquationFit ParametersLor.0 (eV) Lor.5 (cm−1)      ɛ    n_Harmonic    =                    A        n            ⁢              Br        n            ⁢              E        n                            E        n        2            -              E        2            +                        1          /          4                ⁢                  Br          n          2                    -                        iBr          n                ⁢        E            Ampn = An (dimensionless),Enn = En (eV), Brn = Brn (eV)Ampn = An (dimensionless),En1 = En (cm−1), Br1 = Brn (cm−1) Lor.1 (eV) Lor.6 (cm−1)      ɛ    n_Harmonic    =                    A        n            ⁢              E        n                            E        n        2            -              E        2            +                        1          /          4                ⁢                  Br          n          2                    -                        iBr          n                ⁢        E            Ampn = An (eV),Enn = En (eV), Brn = Brn (eV)Ampn = An (cm−1),Enn − En (cm−1), Brn = Brn (cm−1) Lor.2 (eV) Lor.7 (cm−1)      ɛ    n_Harmonic    =                    A        n            ⁢              E        n        2                            E        n        2            -              E        2            +                        1          /          4                ⁢                  Br          n          2                    -                        iBr          n                ⁢        E            Ampn = An (dimensionless),Enn = En (eV), Brn = Brn (eV)Ampn = An (dimensionless),Enn = En (cm−1), Brn = Brn (cm−1)Gaussian:StyleEquationFit Parameters                                                        ɛ              n_Gaussian                        =                                          ɛ                n1                            +                              i                ⁢                                                                  ⁢                                  ɛ                  n2                                                              ,          where                                                                        ɛ              n1                        =                                          2                π                            ⁢              P              ⁢                                                ∫                                      R                    g                                    ∞                                ⁢                                                                                                    ξɛ                        n2                                            ⁡                                              (                        ξ                        )                                                                                                            ξ                        2                                            -                                              E                        2                                                                              ⁢                                      ⅆ                    ξ                                                                                ,                                              using          ⁢                                          ⁢                      ɛ            n2                    ⁢                                          ⁢          as          ⁢                                          ⁢          defined          ⁢                                          ⁢                      below            .                                  Gau.0 (eV) Gau.5 (cm−1)      ɛ    n2    =                    A        n            ⁢              ⅇ                  -                                    (                                                E                  -                                      E                    n                                                                    Br                  n                                            )                        2                                +                  A        n            ⁢              ⅇ                  -                                    (                                                E                  +                                      E                    n                                                                    Br                  n                                            )                        2                              Ampn = An (dimensionless),Enn = En (eV), Brn = Brn = Brn (eV)Ampn = An (dimensionless),En1 = En (cm−1), Br1 = Brn (cm−1) Gau.1 (eV) Gau.6 (cm−1)      ɛ    n2    =                              A          n                          Br          n                    ⁢              ⅇ                  -                                    (                                                E                  -                                      E                    n                                                                    Br                  n                                            )                        2                                +                            A          n                          Br          n                    ⁢              ⅇ                  -                                    (                                                E                  +                                      E                    n                                                                    Br                  n                                            )                        2                              Ampn = An (eV),Enn = En (eV), Brn = Brn (eV)Ampn = An (cm−1),Enn = En (cm−1), Brn = Brn (cm−1) Gau.2 (eV) Gau.7 (cm−1)      ɛ    n2    =                                          A            n                    ⁢                      E            n                                    Br          n                    ⁢              ⅇ                  -                                    (                                                E                  -                                      E                    n                                                                    Br                  n                                            )                        2                                +                                        A            n                    ⁢                      E            n                                    Br          n                    ⁢              ⅇ                  -                                    (                                                E                  +                                      E                    n                                                                    Br                  n                                            )                        2                              Ampn = An (dimensionless),Enn = En (eV), Brn = Brn (eV)Ampn = An (dimensionless),Enn = En (cm−1), Brn = Brn (cm−1)Gauss-Lorentz:                                                        ɛ              n                        =                          i              ⁢                                                          ⁢                                                Φ                  n                                ⁡                                  [                                                                                    ∫                        0                        ∞                                            ⁢                                                                        ⅇ                                                                                    ⅈ                              ⁡                                                              (                                                                  ℏω                                  -                                                                      E                                    n                                                                    +                                                                                                            iBr                                      n                                                                        ⁢                                                                                                                  γ                                        n                                                                            ⁡                                                                              (                                        s                                        )                                                                                                                                                                            )                                                                                      ⁢                            s                                                                          ⁢                                                  ⅆ                          s                                                                                      +                                                                  ∫                        0                        ∞                                            ⁢                                                                        ⅇ                                                                                    ⅈ                              ⁡                                                              (                                                                  ℏω                                  +                                                                      E                                    n                                                                    +                                                                                                            iBr                                      n                                                                        ⁢                                                                                                                  γ                                        n                                                                            ⁡                                                                              (                                        s                                        )                                                                                                                                                                            )                                                                                      ⁢                            s                                                                          ⁢                                                  ⅆ                          s                                                                                                      ]                                                              ,                                                                        where              ⁢                                                          ⁢                                                γ                  n                                ⁡                                  (                  s                  )                                                      =                                          Γ                n                            +                              2                ⁢                                  σ                  n                  2                                ⁢                s                                              ,                                    ⅇ                              Bmix                n                                      =                                          Γ                n                                            σ                n                                              ,          and                                              Φ          ⁢                                          ⁢          is          ⁢                                          ⁢          defined          ⁢                                          ⁢                      below            .                                  StyleΦFit ParametersCalculated ParametersG-L.0 (eV)Φ = AnAn = Ampn (dimensionless),Γn = Blorn (eV),En = Enn (eV), Brn − Brn (eV),σn = Bgaussn (eV)Bmixn = Bmixn (dimensionless)G-L.5 (cm−1)An = Ampn (dimensionless),Γn = Blorn (cm−1),En = Enn (cm−1), Brn = Brn (cm−1)σn = Bgaussn (cm−1)Bmixn = Bmixn (dimensionless) G-L.1 (eV)  G-L.6 (cm−1)  Φ  =            A      n              Br      nn      An = Ampn (eV),En = Enn (eV), Brn = Brn (eV)Bmixn = Bmixn (dimensionless)An = Ampn (cm−1),En = Enn (cm−1), Brn = Brn (cm−1)Bmixn = Bmixn (dimensionless)Γn = Blorn (eV),σn = bgaussn (eV) Γn = Blorn (cm−1),σn = Bgaussn (cm−1) G-L.2 (eV)  G-L.7 (cm−1)  Φ  =                    A        n            ⁢              E        n                    Br      nn      An = Ampn (eV),En = Enn (eV), Brn = Brn (eV)Bmixn = Bmixn (dimensionless)An = Ampn (cm−1),En = Enn (cm−1), Brn = Brn (cm−1)Bmixn = Bmixn (dimensionless)Γn = Blorn (eV),σn = bgaussn (eV) Γn = Blorn (cm−1),σn = Bgaussn (cm−1)Tauc-Lorentz & Egap Tauc-Lorentz:StyleEquationFit ParametersT-L.0 (eV)            ɛ              n_T        -        L              =                  ɛ        n1            +              i        ⁢                                  ⁢                  ɛ          n2                      ,  whereAmpn = An (dimensionless),Enn = Eon (eV),Cn = Cn (eV)Egn = Egn (eV) EgT-L.o (eV)                                          ɛ            n2                    =                                                                                          A                    n                                    ⁡                                      (                                          E                      -                                              Eg                        n                                                              )                                                  2                                                              (                                                            E                      2                                        -                                          Eo                      n                      2                                                        )                                +                                  C                  n                  2                                                      ⁢                                                  ⁢                                          Θ                ⁡                                  (                                      E                    -                                          Eg                      n                                                        )                                            E                                                          and                                                  ɛ            n1                    =                                    2              π                        ⁢            P            ⁢                                          ∫                                  R                  g                                ∞                            ⁢                                                                                          ξɛ                      n2                                        ⁡                                          (                      ξ                      )                                                                                                  ξ                      2                                        -                                          E                      2                                                                      ⁢                                  ⅆ                  ξ                                *                                                           Egap T-L.0: same asabove, exceptEgap = Eg (eV)Ionic1 & Ionic2:StyleEquationFit ParametersIon1.0 (eV)  Ion1.5 (cm−1)      ɛ    n_Ion1    =            ɛ      ∞n        +                            E          Tn          2                ⁡                  (                                    ɛ              dcn                        -                          ɛ              ∞n                                )                                      E          Tn          2                -                  E          2                -                              iBr            n                    ⁢          E                    edcn= εdcn (dimensionless),einfn = ε∞n (dimensionless),Eton = ETn (eV), Brn = Brn(eV)edcn = εdcn (dimensionless),einfn = ε∞n (dimensionless),Eton = ETn (cm−1), Brn = Brn (cm−1) Ion2.0 (cV)  Ion2.5 (cm−1)      ɛ    n_Ion1    =            ɛ      dcn        (                            E          Tn          2                          E          Ln          2                    +                                    E            Tn            2                    ⁡                      (                          1              -                                                E                  Tn                  2                                                  E                  Ln                  2                                                      )                                                E            Tn            2                    -                      E            2                    -                                    iBr              n                        ⁢            E                                )  edcn= εdcn (dimensionless),eton = ETn (eV), Brn = Brn (eV)Elon = El,n (eV)edcn = εdcn (dimensionless),Eton = ETn (cm−1), Brn = Brn (cm−1)Elon = ELn (eV)TOLO:StyleEquationFit ParametersTOLO.0 (eV)  TOLO.5 (cm−1)      ɛ    n_TOLO    =            A      n        ⁢                            E          lon          2                -                  E          2                -                              iB                          l              ⁢              on                                ⁢          E                                      E          ton          2                -                  E          2                -                              iB            ton                    ⁢          E                    Ampn = An (dimensionless),Elon = Elon (eV), Eton = Eton (eV)Blon = Blon (eV), Bton = Bton (eV)Ampn = An (dimensionless),Elon = Elon (cm−1), Eton = Eton (cm−1)Blon = Blon (cm−1), Bton = Bton (cm−1) TOLO.1 (eV)  TOLO.6 (cm−1)      ɛ    n_TOLO    =                    A        n                    B        ton              ⁢                            E          lon          2                -                  E          2                -                              iB            lon                    ⁢          E                                      E          ton          2                -                  E          2                -                              iB            ton                    ⁢          E                    Ampn = An (eV),Elon = Elon (eV), Eton = Eton (eV)Blon = Blon (eV), Bton = Bton (eV)Ampn = An (cm−1),Elon = Elon (cm−1), Eton − Eton (cm−1)Blon = Bton (cm−1), Bton − Bton (cm−1) TOLO.2 (eV)  TOLO.7 (cm−1)      ɛ    n_TOLO    =            A      n        ⁢                  E        ton                    B        ton              ⁢                            E          lon          2                -                  E          2                -                              iB            lon                    ⁢          E                                      E          ton          2                -                  E          2                -                              iB            ton                    ⁢          E                    Ampn = An (dimensionless),Elon = Elon (eV), Eton = Eton (eV)Blon = Blon (eV), Bton = Bton (eV)Ampn = An (dimensionless),Elon = Elon (cm−1), Eton − Eton (cm−1)Blon = Blon (cm−1), Bton = Bton = (cm−1)GLAD:Separationn = Splitn × BrnAsymn > 0: (lower-energy peak) < (higher-energy peak)Asymn < 0: (lower-energy peak) > (higher-energy peak)Asymn = 0: (lower-energy peak) = (higher-energy peak)The two peaks are Gauss-Lorentz oscillators and are defined as                                                         ɛ              n                        =                          i              ⁢                                                          ⁢                                                Φ                  n                                ⁡                                  [                                                                                    ∫                        0                        ∞                                            ⁢                                                                        ⅇ                                                                                    ⅈ                              ⁡                                                              (                                                                  ℏω                                  -                                                                      E                                    n                                                                    +                                                                                                            iBr                                      n                                                                        ⁢                                                                                                                  γ                                        n                                                                            ⁡                                                                              (                                        s                                        )                                                                                                                                                                            )                                                                                      ⁢                            s                                                                          ⁢                                                  ⅆ                          s                                                                                      +                                                                  ∫                        0                        ∞                                            ⁢                                                                        ⅇ                                                                                    ⅈ                              ⁡                                                              (                                                                  ℏω                                  +                                                                      E                                    n                                                                    +                                                                                                            iBr                                      n                                                                        ⁢                                                                                                                  γ                                        n                                                                            ⁡                                                                              (                                        s                                        )                                                                                                                                                                            )                                                                                      ⁢                            s                                                                          ⁢                                                  ⅆ                          s                                                                                                      ]                                                              ,                                                                        where              ⁢                                                          ⁢                                                γ                  n                                ⁡                                  (                  s                  )                                                      =                                          Γ                n                            +                              2                ⁢                                  σ                  n                  2                                ⁢                s                                              ,                                    ⅇ                              Bmix                n                                      =                                          Γ                n                                            σ                n                                              ,          and                                              Φ          ⁢                                          ⁢          is          ⁢                                          ⁢          defined          ⁢                                          ⁢                      below            .                                  StyleΦFit ParametersGLAD.0 (eV)Φ = AnAn = Ampn (dimensionless),En = Enn (eV), Brn = Brn (eV),Splitn = Splitn, Asymn = Asymn, Bmixn = Bmixn(Splitn, Asymn, & Bmixn all dimensionless)GLAD.5 (cm−1)An = Ampn (dimensionless),En = Enn (cm−1), Brn = Brn (cm−1),Splitn = Splitn, Asymn = Asymn, Bmixn = Bmixn(Splitn, Asymn, & Bmixn all dimensionless) GLAD.1 (eV)   GLAD.6 (cm−1)  Φ  =            A      n              Br      nn      An = Ampn (eV),En = Enn (eV), Brn = Brn (eV)Splitn = Splitn, Asymn = Asymn, Bmixn = Bmixn(Splitn, Asymn, & Bmixn all dimensionless)An − Ampn (cm−1),En = Enn (cm−1), Brn = Brn (cm−1)Splitn = Splitn, Asymn = Asymn, Bmixn = Bmixn(Splitn, Asymn, & Bmixn all dimensionless) GLAD.2 (eV)   GLAD.7 (cm−1)  Φ  =                    A        n            ⁢              E        n                    Br      nn      An = Ampn (dimensionless),En = Enn (eV), Brn = Brn (eV)Splitn − Splitn, Asymn = − Asymn, Bmixn − Bmixn(Splitn, Asymn, & Bmixn all dimensionless)An = Ampn (dimensionless),En Enn (cm−1), Brn = Brn (cm−1)Splitn = Splitn, Asymn = Bmixn = Bmixn(Splitn, Asymn, & Bmixn all dimensionless)CPPB:                                                                        ɛ                n                            =                                                                    Φⅇ                                          ⅈθ                      n                                                        ⁡                                      (                                          Γ                                                                        2                          ⁢                                                      E                            gn                                                                          -                                                  2                          ⁢                          E                                                -                                                  i                          ⁢                                                                                                          ⁢                                                      Γ                            n                                                                                                                )                                                                    μ                  n                                                      ,                                          μ                n                            =                                                ±                  1                                /                2                                                                                                                    ɛ                n                            =                                                Φⅇ                                      ⅈ                    ⁢                                                                                  ⁢                                          θ                      n                                                                      ⁢                                  ln                  ⁡                                      (                                                                  2                        ⁢                                                  E                          gn                                                                    -                                              2                        ⁢                        E                                            -                                              i                        ⁢                                                                                                  ⁢                                                  Γ                          n                                                                                      )                                                                        ,                                          for                ⁢                                                                  ⁢                                  μ                  n                                            =              0                                                                        where            ⁢                                                  ⁢            Φ            ⁢                                                  ⁢            is            ⁢                                                  ⁢            defined            ⁢                                                  ⁢                          below              .                                                   StyleΦFit ParametersCPPB.0 (eV)Φ = AnAmpn = An (dimensionless), Enn = Egn (eV), Brn = Γn (eV),mun − μn − −½, , 0, ½; Phasen = θn (dimensionless)CPPB.5 (cm−1)Ampn = An (dimensionless), Enn = Egn (cm−1), Brn = Γn (cm−1)mun = μn = −½, , 0, ½; Phasen = θn (dimensionless) CPPB.1 (eV) CPPB.6 (cm−1)  Φ  =            A      n              Γ      n      Ampn = An (eV), Enn = Egn (eV), Brn = Γn (eV),mun = μn = −½, , 0, ½; Phasen = θn (dimensionless)Ampn = An (cm−1), Enn = Egn (cm−1), Brn = Γn (cm−1)mun = μn = −½, , 0, ½; Phasen = θn (dimensionless) CPPB.2 (eV) CPPB.7 (cm−1)  Φ  =                    A        n            ⁢              E        n                    Γ      nn      Ampn = An (dimensionless), Enn = Egn (eV), Brn = Γn (eV),mum = μn = −½, , 0, ½; Phasen = θn (dimensionless)Ampn = An (dimensionless), Enn = Egn (cm−1), Brn = Γn (cm−1)mun = μn = −½, , 0, ½; Phasen = θn (dimensionless)Tguy:            ɛ      Tanguy_n        =                                        A            n                    ⁢                                    R              n                                                            (                          E              +                              i                ⁢                                                                  ⁢                                  Γ                  n                                                      )                    2                    ⁢              {                                            g              u                        ⁡                          (                              ξ                ⁡                                  (                                      E                    +                                          i                      ⁢                                                                                          ⁢                                              Γ                        n                                                                              )                                            )                                +                                    g              u                        ⁡                          (                              ξ                ⁡                                  (                                                            -                      E                                        -                                          i                      ⁢                                                                                          ⁢                                              Γ                        n                                                                              )                                            )                                -                      2            ⁢                                          g                u                            ⁡                              (                                  ξ                  ⁡                                      (                    0                    )                                                  )                                                    }              , where                                                            g              a                        ⁡                          (              ξ              )                                =                                    2              ⁢              ln              ⁢                                                          ⁢              ξ                        -                          2              ⁢              π              ⁢                                                          ⁢              c              ⁢                                                          ⁢                              o                (                                  π                  ⁢                                                                          ⁢                  ξ                                )                                      -                          2              ⁢                                                          ⁢                              ψ                ⁡                                  (                  ξ                  )                                                      -                          1              /              ξ                                                                                              ξ              ⁡                              (                z                )                                      =                                                            R                  a                                                                      E                                                                  x                        -                                            ⁢                      a                                                        -                  z                                                              ,          and                                                          ψ            ⁡                          (              z              )                                =                                                    ⅆ                                  (                                      ln                    ⁢                                                                                  ⁢                                          Γ                      ⁡                                              (                        z                        )                                                                              )                                                            ⅆ                z                                      ⁢                          (                              digamma                ⁢                                                                  ⁢                function                            )                                           StyleFit ParametersTguy.0 (eV)Ampn = An (eV2),Egn = Eg,n (eV),Bn = Γn (eV),Rn = Rn (eV)Ingn =Psemi-E0 [only PS-E0.0 (eV) Style available]:Fit ParameterDescriptionDimensionsAmpAmplitude of ε2 at EgapAmpn = Ampn (dimensionless)EgapBandgap EnergyEgapn = Egapn (eV)BrBroadening (Gaussian type)*Brn = Brn (eV)EwidWidth of absorption regionEwidn = Ewidn (eV)(Egap to high energy cut-off)MposConnecting point of polynomials,Mposn = Mposn(as a fraction of Ewid),(dimensionless)measured from (Egap + Ewid)*MampAmplitude at connecting pointMampn = Mampnof polynomials (as a fraction of(dimensionless)Amp)*O2ndSecond order polynomial factors*02ndn = 02nd (dimensionless)
For additional insight see the J.A. Woollam Co. WVASE32 (Registered Trademark), Manual which describes the GENOSC™ Layer. Said J.A. Woollam Co. WVASE32 Manual is incorporated by reference herein. Also note that insight to the application of the Drude Model with mean scatteing time and resistivity as parameters is given in “Introduction to Solid Stae Physics”, Kittel, 6th Ed, John Wiley & Sons, (1986), P. 257. More insight to the Tauc-Lorentz Model can be found in “Parameterization of the Optical Functions of Amorphous Materials in the Interband Region”, Jellison & Modine, Appl. Phys. Lett. 69, 371 (1996). More insight to the Gauss-Lorentz Model can be found in an article by Kim et al., at Phys. Rev. B 45, 11749 (1992). More information about the CPPB Critical Point Parabolic Band Model can be found in “Modulation Spectroscopy/Electric Field Effects on the Dielectric Function of Semiconductors”, Handbook of Physics, Vol. 2 Editor Balkanski, North Holland Pages 125-127, (1980). More Information about CPM0, CPM1, CPm2 and CPm3 Models can be found in “Other Dispersion Relations for GaP, GAAS, GaSb, InP InAs, InSb Alx Ga1-x, GaxAsyP1-y”, J. Appl. Phys. 66, 6030 (1989). More Information about the Tangay Model can be found in “Optical Dispersion of Wannier Excitons”, Phys. Rev. Lett. 75, 4090 (1995) and Errata, Phys. Rev Lett. 76, 716 (1996).
Continuing, insight to Special Properties, Usefulness and what Materials some of said Oscillator Structures are applied to are: